Bayesian seasonal adjustment modeling. The model allows for a decomposition of a time series into trend, seasonal, and an error component.

Routine BAY_SEA is based upon the algorithm published by Akaike (1980). This algorithm uses a Bayesian approach to the problem of fitting the following autoregressive model for a time series Wt decomposed into a trend and a seasonal component.

Adopting the notation described earlier in the Usage Notes section of this chapter, if

then a seasonal autoregressive model can be represented by the following relationship:

where Wt is the stationary time series with mean μ, Tt denotes an underlying trend, St seasonal component and At a noise or irregular component.

A non‑Bayesian approach to this problem would be to estimate the trend and seasonal components by minimizing

where p is the period of the seasonal component, and d, r, and z are properly chosen constants.

In BAY_SEA, the approach is to select the parameter d, which controls the smoothness of the trend and seasonality estimates, using Bayesian methods. The prior distribution controls the smoothness of the trend and seasonal components by assuming low order Gaussian autoregressive models for some differences of these components. The choice of the variance of the Gaussian distribution is realized by maximizing the log likelihood of the Bayesian model.

The other smoothing parameters, r and z, are determined by the value of RIGID. The default value for RIGID is 1. Larger values of RIGID produce a more rigid seasonal pattern. Normally, a series is first fit using the default value for RIGID. The smoothness of the trend and seasonality estimates are examined and then RIGID is either increased or decreased depending upon whether more or less seasonal smoothing is needed.

Additionally, BAY_SEA selects the optimum autoregressive model as the model that minimizes ABIC.

where the likelihood in this case is the mixed Bayesian maximum likelihood. Smaller values of ABIC represent a better fit.

This example uses unadjusted unemployment for women over 20 years of age in the U.S. for 1991‑2001, as reported by the U.S. Bureau of Labor Statistics (www.bls.gov). Figure 18 displays this data together with the smoothed preditions from BAY_SEA. Figure 19 displays the same data with trend predictions from BAY_SEA.